Additive Measures

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton

We have now seen three paradoxes whose resolutions have been postponed. They are Zeno's paradox of the stadium, in Davy's version; Zeno's paradox of measure; and the geometric version of Aristotle's wheel. They each concern the notion of measure, that is, the size of things. Their resolution will reside in identifying the need to separate two things that have been tacitly coupled: the size of things and the sizes of the points that compose them.

Consider some system consisting of infinitely many points, such as shown on the left in the figure below. We will no longer assume that a count of the points fixes magnitudes associated with the system. The count will not fix the volume of the system (shown shaded in the middle). It will not fix the distances between the points (shown on the right).

It proves to be a delicate matter to effect this separation consistently. The separation requires ideas about infinity from the earlier chapter on infinite sets.

Measure theory is the applicable, formal theory of the sizes of things. Once we have a clearer grip on measures, we can see that all three paradoxes depend on the tacit assumption that the points of an infinite magnitudes fix their measures. In the paradoxical cases, this proves to be a false assumption that generates the paradoxes.

The key idea to be developed in this chapter concerns the relationship between an extended magnitude and its parts.
• If there are finitely many parts, then magnitude of the whole is just the sum of the magnitude of the parts.
• If there is a countable infinity of parts, then magnitude of the whole is still the sum of the magnitude of the parts.
• If there is an uncountable infinity of parts, then the magnitude of the whole is decoupled from magnitude of the parts. For there is no way to sum uncountably many component magnitudes.

This modern resolution of the paradoxes of measure requires a nineteenth century understanding of the different orders of infinity. It can only arise once we can distinguish a countable infinity from an uncountable infinity.

The idea of an additive measure

Extensive magnitudes are found throughout mathematics and the sciences. They are quantities that increase in proportion to the size of the system considered. The notion of an additive measure provides a general account of these extensive magnitudes, applicable to many cases.

The general notion applies to systems composed of many points. Its defining characteristics are:

(Additive Measure) For subsets of points defining a magnitude:

1. (non-negativity) The measure assigns a non negative real number to some subsets.
2. (null set) The measure assigns zero to the empty set.
3. (additivity) If two subsets are disjoint, then the measure of their union is sum of the measures of their two parts.

The simplest additive measure is the counting measure routinely employed for systems consisting of finitely many points. For example, we measure how many eggs we have by counting them; and when we combine our eggs the counted numbers add up.

A dozen eggs combined with half a dozen eggs gives us 18 eggs = 12 eggs + 6 eggs.

We find additive measures in many more places. They are lengths, areas and volumes and times elapsed. They apply to the masses of bodies, to energies and to momenta. In the figure below, they are applied to simple planar areas. We have an area whose parts individually have areas 1, 3 and 2. Then the combined area is 1 + 3 + 2 = 6.

Finite Additivity

It is routinely assumed that this rule allows us to add the measures of finitely many parts:

(Finite additivity)
If a system has finitely many disjoint parts, part-1, part-2, ..., part-n, then:

measure (combined parts)
= measure (part-1) + measure (part-2) + ... + measure (part-n)

This rule is justified by breaking up the summation into n-1 summations of two parts. That is, we first sum the measures of the first two parts, and then the next two and so on until all n-1 summations are completed.

Here is how it looks spelled out in greater detail.

measure (part-1) + measure (part-2) + ... + measure (part-n)

= [measure (part-1) + measure (part-2) ] + ... + measure (part-n)

= measure (parts 1,2) + measure (part-3) + ... + measure (part-n)

= [ measure (parts 1,2) + measure (part-3) ] + ... + measure (part-n)

= measure (parts 1,2, 3) + measure (part-3) + ... + measure (part-n)
etc.

Since arithmetic addition is associative, the order in which these additions are carried out does not matter. We will always arrive at the same sum.

Countable Additivity

A new rule appears when we seek to add measures in systems that have infinitely many parts. The key condition is that the number of parts to be added is countably infinite, that is, of size of the set of natural numbers in the sense of Cantor's theory, ℵ₀. For then we can number all the parts whose measures are to be added:

and so on infinitely. There is a rule for adding the measures of these parts, called the rule of "countable additivity."

(countable additivity)
If a system has finitely many disjoint parts, part-1, part-2, ..., part-n, then:
         measure (combined parts)
              = measure (part-1)
                     + measure (part-2)
                            + ... (and so on infinitely)

At first glance, this rule looks little different from the rule of finite additivity. However, there is an important difference. We could justify the rule of finite additivity for n parts just by writing down a calculation with n-1 pairwise summations. This procedure fails for the case of countable additivity. No matter how many additions we carry out finitely, we will never have completed the infinitely many additions called for in the summation.

The solution is to introduce as a definition what it is to sum infinitely many measures; and to do it in a way that naturally extends the rule of finite additivity. We form the infinite sequence of partial sums:

measure (part-1)
measure (part-1 and part-2)
measure (part-1 and part-2 and part-3)
...

How can "all goes well" fail? Since each measure is non-negative, the partial sums cannot decrease as we form them. All that can go wrong is that the sum is infinite. "Infinity" is not a real number, so it lies outside the definition. It is customary, however, to overlook this complication and call such a result an "infinite measure."

Each of these can be formed using finite additivity. What will result is a sequence of numbers. If all goes well, that sequence will converge to some definite number and that number is the infinite sum.

We have already seen an example of this procedure in Zeno's dichotomy paradox. To complete the course the runner had to traverse a distance of 1/2, and then 1/4, and then 1/8, and so on infinitely. We had to add these infinitely many distances.

We saw there that the partial sums in this infinite sequence of finite summations are:

1/2 = 1 - 1/2

1/2 + 1/4 = 3/4 = 1 - 1/4

1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8

1/2 + 1/4 + 1/8 + 1/16
= 15/16 = 1 - 1/16

No matter how many more of these sums we form, we are always some finite away from 1. We might add the first ten terms in the sequence and have

Since these sums get arbitrarily close to one, it is quite natural to declare as a definition that the infinite sum is one:

Must we accept countable additivity?

Countable additivity is a perfectly well defined mathematical operation. If one has a magnitude with a countable infinity of parts, it would seem perverse not to use it. We cannot complain, for example, that summing an infinity of terms is inadmissible since completing an infinity of actions is objectionable. We have seen through our investigation of supertasks that this objection fails.

There are many cases in which we do want to accept countable additivity. One example we have already seen is Zeno's paradox of the dichotomy. In the stages of the paradox, the runner passes over a countable infinity of distances, 1/2, 1/4, 1/8, ...

Perverse or not, nothing compels us to add the rule of countable additivity to the rule of finite additivity. We can choose to add it or not as seems appropriate to the case at hand. For the two operations of finite addition and countably infinite addition are distinct operations. No logical contradiction arises if we apply the first, but not the second.

What does it look like to retain finite additivity but not countable additivity? Here is a simple example. Imagine that we have a system consisting of a countable infinity of parts of measure

We can use finite additivity to compute the measure of any finite combination of these parts.

However, without invoking the rule of countable addivity, no summation will lead to the measure of the full system. For no matter how we try to arrive at that measure, we will have to add a countable infinity of measures of parts.

The rule of countable additivity would tell us that the measure of the whole is just 1. However without that rule applying, nothing stops us assigning a different measure to the whole. We might assign, say, a measure of 2. It would be an odd thing to do since no finite sum of measures of parts exceeds 1. However there is no logical contradiction in the assignment.

The most notable case and the one that will be important for us here arises in the case of systems composed of a countable infinity of points and where each point is measure zero.

measure (point-1) = 0
measure (point-2) = 0
measure (point-3) = 0
etc.

Then the rule of countable additivity gives us:

We might be tempted to represent lengths by sets of rational numbers. That is, the interval of lengths from 0 to 1 would be represented by the set of rational numbers between 0 and 1, where each individual rational number is assigned a measure zero. It is tempting to use this representation since, as we saw earlier, the rational numbers are dense in the interval 0 to 1.

is the escape to assign a very tiny measure ε>0 to each rational? Then we run into the opposite problem. The measures of the infinitely many rationals in the interval from 0 to 1 would sum to infinity.

This choice would be a quite troublesome, since the rational numbers form a countable set. If we applied the rule of countable additivity to the measures of the rational numbers in this interval, we would find that they sum to zero. We would have a degenerate notion of lengths: all intervals, would have zero length.

In a later chapter, we shall see another case in which it is appealing to abandon countable additivity. It arises in probability theory when we deal with the problem of the infinite lottery.

Uncountable Additivity?

Uncountable sets represent lengths

We have just seen that a set of rational numbers fails to represent lengths made up of a dense set of points if we want these lengths to have non-trivial, countably additive measures (that is, measures other than zero and infinity). The standard solution is to represent lengths (and other similar magnitudes) by sets of continuum size of real numbers.

Take again the interval from 0 to 1. It is represented by the set of all real numbers lying between 0 and 1. Each real number represents a point in the line segment. Each real number is assigned zero measure.

This much seems to set us up for the same trouble that faced a representation of lengths by sets of rational numbers. For we have points--real numbers--of zero measure; and an infinity of them taken together gives us the line from 0 to 1. The only difference now is that the reals form an infinite set of a larger size, an uncountable set that is the size of the continuum.

All that is needed to arrive at this same trouble is for there to be a way to add the measures of the uncountably many points forming the continuum between zero and one. The rule of countable additivity can be applied, but it falls far short of what is needed. At best it adds up the measures of a countable infinity of real number points. We would have a sequence of points like:

Note: these are individual points on the number line, each of which we assume to have zero measure. These are not lengths.

measure (point-1)
measure (point-1 and point-2)
measure (point-1 and point-2 and point-3)
...

The ensuing, limiting sum will be defined. Applied to points in the interval of reals from 0 to 1, it can tell us us that the measure of the resulting countably infinite set is zero in size. It is just the familiar countably infinite sum:

That zero sum is untroubling since any set of points of countable size is a negligible portion of the full set of real numbered points between zero and one.

This last example mentioned rational numbers. We can also select countably infinite sets of irrational numbers in [0,1], such as:

What we need is a notion of uncountable additivity that would allow us to sum the uncountably many zero measures of all the points. We would have a sequence like:

There is no way to form such a sequence for a set of points of continuum size. Even to try to write it as a list like this is misleading. For a list suggests that we have a countable set with a first member, a second member, a third member, and so on. A continuum-sized set is much, much, much bigger and admits of no enumeration. Efforts to use the sort of limit approach that worked for countable additivity have not produced useful results. Instead...

This is a "light-bulb" moment in which we see the key fact!

To be more precise, it is possible to define notions of uncountable additivity. They have not proved to be useful in practical applications. Even if someone defines a notion of uncountable additivity, nothing compels us to accept it in applications that interest us.

This is the most important fact of this chapter, as far as resolving the paradoxes of measure are concerned. When the underlying sets are continuum sized, measure theory separates the size of the set from the measure assigned.

One set, many measures

The lack of a rule of uncountable additivity opens a gap between two types of measures: the zero measures of the individual points and the measures that can be assigned to intervals such as extend from 0 to 1. Because of this gap, we have great latitude in the measures that we can assign to systems consisting of continuum many points.

In the case of the interval from 0 to 1, a common choice is a uniform measure, officially called the "Lesbegue measure." This measure is assigned to infinite subsets of the interval of continuum size. The rule is just this:

The whole interval [0,1] is assigned measure 1-0 = 1.
The half interval [0,0.5] is assigned measure 0.5-0 = 0.5.
The interval [0.15, 0.45] is assigned measure 0.45-0.15 = 0.30.
etc.

What matters is that the totality of these assignments are consistent. We can check this is so for a few simple cases.

The rule of finite additivity holds. For example, the intervals [a, b] and [b,c] combine to form the interval [a,c]. The measures of the first two are (b-a) and (c-b). Their sum is (c-b) + (b-a) = (c-a), which is the measure of the interval [a,c].

This Lebesgue measure conforms with the rule of countable additivity. Take the infinitely many intervals that arise in the dichotomy:

While this Lebesgue measure is the standard one used, nothing compels it. We can replace it with other measures that conform with finite and countable additivity, while still preserving the measure zero of the individual points.

A simple alternative measure has values that are twice those of the Lebesgue measure. That is:

Then everything proceeds as before, but now all the measures are doubled. Since the first measure produced no contradiction, the same will be true of the doubled measure. We have:

The whole interval [0,1] is assigned measure 2-0 = 2.
The half interval [0,0.5] is assigned measure 1-0 = 1.
The interval [0.15, 0.45] is assigned measure 0.90-0.30 = 0.60.
etc.

We can reproduce the example of the dichotomy, now with doubled measures. The intervals

Both these measure--the Lebesgue measure and the doubled Lebesgue measure--are uniform measures. They are just the first two of infinitely many measures that can be assigned to the interval of reals from 0 to 1. Nothing requires the measure to be uniform. We can have measures that assign greater values to intervals close to zero; or close to 1; or elsewhere; and so on in innumerable variations.

For calculus experts: These further measures assign a non-negative number to each interval of reals and all their unions. The key condition to be met is that all the numbers assigned are consistent with one another. This condition is routinely met by specifying the measure through a density function, ρ. It assigns real numbers to each real in the interval. The measure of an interval is then just the integral of the density function over that interval.

Induced measures

The discussion of the last few sections shows that there is no, simple assured relationship between the measures assigned to individual points forming some system and the measures assigned to the system as a whole.

We have cases in which the measures assigned to the individual points fix the measures assigned to the whole. That is, these are cases in which the measures on the whole are induced by the measures on the individual points.

One is the case of the counting measure. If we have two cartons of eggs, then our total egg count is sum of the counts of eggs in the two cartons.

Another, richer example is the case of the rationals between zero and one. Once we assign a zero measure to each rational individually, countable additivity forces us to assign a zero measure to the whole interval.

Alternatively, the measures on the individual points can fail to fix the measure on the whole. The notable case is the one explored in the last section. We assign zero measure to the individual points of an interval of the real continuum. We remain free to assign many different measures to whole interval.

The Paradoxes

The three paradoxes listed at the start of this chapter depend essentially on neglecting the possibility that the measure on the whole may not be fixed by the measure on the parts. That is:

In sets of continuum size, the zero measure of the individual points does not determine the measure of the totality.

Zeno's paradox of measure, again

The paradox considered an infinitely divisible extended magnitude dissolved into its infinity of dimensionless parts.

Each part has zero measure. It was assumed that the measure of the totality could be recovered by adding up the infinitely many zeroes to get zero:

If the magnitude consists of a countable infinity of points, such as a set of rational numbers, then this addition is authorized by the rule of countable additivity. Then the measures on the points does fix the measure on the totality.

However the extended magnitudes used in geometry and elsewhere are composed of an uncountable set of continuum many points. Standard theory provides no notion of uncountable additivity such that this last summation can proceed. At best the summation can extend only over a countable subset of all the points. It can reassure us that the subset of rational numbers in some interval is a measure zero set. But it can do no more.

The summation violates the conclusion above that the measure of the whole is not fixed by those of the individual points.

Zeno's paradox of the stadium

In Davy's version of the paradox, we we have three bodies, AA', BB' and CC' of equal size, consisting of extended magnitudes of infinitely many points.. AA' is at rest and bodies BB' and CC' move past in opposite directions, such that point B' point sweeps past half of body AA', but the full extend of body CC'.

The result is that a one-to-one correspondence is established between half the points of AA' and all the points of CC' (shown on the left). That is compared with the natural one-to-one correspondence between half the points of AA' and CC' (shown on the right). Combining these correspondence, it follows that the points of a full length, CC', can be mapped one-to-one to the points of a half of its length.

The inference then made is that the half interval must have the same magnitude as the full interval CC'. If we use these magnitudes to measure the time of the motions depicted, we end up with Aristotle's paradoxical conclusion "half the time is equal to its double."

This inference, we now see, is a fallacy. The two intervals have the same number of points and every one of them is of zero measure. Since the number of points is continuum-sized, measure theory has separated the size of the sets of points from the measure assigned. We can and do, without contradiction, assign the full measure to the full length CC' but only half the measure to half of the length.

Aristotle's wheel

The geometric version of the Aristotle wheel paradox depended on identifying a one-to-one correspondence between the infinitely many points of the circumference of a smaller circle and the infinitely many points of a larger circle.

The same fallacy is then committed. Since the two circumferences are composed of the same number of points, it is inferred fallaciously that they are of the same measure, that is, the same length. That then contradicts the Euclidean result that the larger circle has a larger circumference.

Since measure theory has separated the size of the underlying set from the measure assigned in this case of continuum sized sets of points, we are free to assign different measures to each circle. That is what our Euclidean geometry does. It assigns a circumference of π times the circle's diameter to each circle. It follows that, under this assignment, the larger circle has the larger circumference.

Non-measurable Sets

We have now seen enough measure theory to be able to resolve the three paradoxes listed at the start of the chapter. There is an important aspect of measure theory that should be mentioned here.

In systems of continuum size, only some subsets can consistently be assigned a countably additive measure. This is a universal problem. The Lebesgue measure defined above, for example, can be extended from the intervals of the original definition to many more subsets by finite and countable additions. However there remain many more subsets for which no extension of the Lebesgue measure is possible.

For this reason, the specification of a measure routinely requires three elements.
• First is the set of points that delimit the system of interest.
• Second is a specification of those subsets to which the measure will be assigned.
•Third is the specification of the value of the measure assigned to each of these subsets.

For more, see "Axiom of Choice" in a later chapter.

The identification of which sets are non-measurable has proven to be delicate. The difficulty is that these sets are, on our best understanding, non-constructible. That means that we can infer that they exist, but we cannot point to a specific set and announce "that set is not measurable!"

The standard example of such a set is the "Vitali set," whose existence is inferred but not displayed. I have elsewhere provided an elementary account it. See a later chapter and also

To Ponder

Consider an account of measure that allows only finite additivity, but not countable additivity. What do we gain? What do we lose?

Is there no way to sum an uncountable infinity of measures? Perhaps we should just try harder?

Consider the three paradoxes resolved in this chapter: Zeno's paradoxes of measure and the stadium and the geometric version Aristotle's wheel. They use notions derived from modern measure theory. Might it be possible to explain to thinkers in antiquity how the paradoxes are resolved in terms compatible with their methods.